Four-Dimensional Algebraic Tori

The study of the birational properties of algebraic $k$-tori began in the sixties and seventies with work of Voskresenkii, Endo, Miyata, Colliot-Th\'el\`ene and Sansuc. There was particular interest in determining the rationality of a given algebraic $k$-tori. As rationality problems for algebraic varieties are in general difficult, it is natural to consider relaxed notions such as stable rationality, or even retract rationality. Work of the above authors and later Saltman in the eighties determined necessary and sufficient conditions to determine when an algebraic torus is stably rational, respectively retract rational in terms of the integral representations of its associated character lattice. An interesting question is to ask whether a stably rational algebraic $k$-torus is always rational. In the general case, there exist examples of non-rational stably rational $k$-varieties. Algebraic $k$-tori of dimension $r$ are classified up to isomorphism by conjugacy classes of finite subgroups of GL$_r(\mathbb{Z})$. This makes it natural to examine the rationality problem for algebraic $k$-tori of small dimensions. In 1967, Voskresenskii proved that all algebraic tori of dimension 2 are rational. In 1990, Kunyavskii determined which algebraic tori of dimension 3 were rational. In 2012, Hoshi and Yamasaki determined which algebraic tori of dimensions 4 and 5 were stably (respectively retract) rational with the aid of GAP. They did not address the rationality question in dimensions 4 and 5. In this paper, we show that all stably rational algebraic $k$-tori of dimension 4 are rational, with the possible exception of 10 undetermined cases which fall into 2 families. We reprove the stable rationality of the exceptional families of algebraic tori non-computationally.